COMPUTER GRAPHICS COURSE Viewing and Projections Georgios - - PowerPoint PPT Presentation

COMPUTER GRAPHICS COURSE

Georgios Papaioannou - 2014

Viewing and Projections

VIEWING TRANSFORMATION

The Virtual Camera

Y

• All graphics pipelines perceive the virtual world

through a virtual observer (camera), also positioned in the 3D environment

“eye” (virtual camera)

Eye Coordinate System (1)

• The virtual camera or “eye” also has its own

coordinate system, the eye coordinate system

Eye coordinate system (ECS) (WCS) eye Global (world) coordinate system X Y Z X Y Y Z

Eye Coordinate System (2)

• Expressing the scene’s geometry in the ECS is a

natural “egocentric” representation of the world:

– It is how we perceive the user’s relationship with the environment – It is usually a more convenient space to perform certain rendering tasks, since it is related to the ordering of the geometry in the final image

Eye Coordinate System (3)

• Coordinates as “seen” from the camera reference

frame

Y X ECS

Eye Coordinate System (4)

• What “egocentric” means in the context of

transformations?

– Whatever transformation produced the camera system  its inverse transformation expresses the world w.r.t. the camera

• Example: If I move the camera “left”, objects appear

to move “right” in the camera frame:

WCS camera motion Eye-space object motion

Moving to Eye Coordinates

• Moving to ECS is a change of coordinates

transformation

• The WCSECS transformation expresses the 3D

environment in the camera coordinate system

• We can define the ECS transformation in two ways:

– A) Invert the transformations we applied to place the camera in a particular pose – B) Explicitly define the coordinate system by placing the camera at a specific location and setting up the camera vectors

WCSECS: Version A (1)

• Let us assume that we have an initial camera at the
• rigin of the WCS
• Then, we can move and rotate the “eye” to any pose

(rigid transformations only: No sense in scaling a camera): 𝐩𝑑, 𝐯, 𝐰, 𝐱 = 𝐒1𝐒2𝐔1𝐒𝟑 … . 𝐔𝑜𝐒𝑛 𝐩, ො 𝐟1, ො 𝐟2, ො 𝐟3

• The eye space coordinates of shapes, given their

WCS coordinates can be simply obtained by: 𝐰𝐹𝐷𝑇 = 𝐍𝑑

−1𝐰𝑋𝐷𝑇

𝐍𝑑

WCSECS: Version A (2)

• This version of the WCSECS transformation

computation is useful in cases where:

– The camera system is dependent on (attached to) some moving geometry (e.g. a driver inside a car) – The camera motion is well-defined by a simple trajectory (e.g. an orbit around an object being inspected)

WCSECS: Version B (“Look At”) (1)

• Let us directly define a camera system by specifying

where the camera is, where does it point to and what is its roll (or usually, its “up” or “right” vector)

right up roll front look-at camera position

WCSECS: Version B (“Look At”) (2)

• The camera coordinate system offset is the eye

(camera) position 𝐩𝑑

• Given the look-at position (the camera target) 𝐪𝑢𝑕𝑢

and 𝐩𝑑, we can determine the “front” direction:

Ԧ 𝐞𝑔𝑠𝑝𝑜𝑢 = 𝐪𝑢𝑕𝑢 − 𝐩𝑑 (normalized)

𝐪𝑢𝑕𝑢 𝐩𝑑

WCSECS: Version B (“Look At”) (3)

• The “up” or “right” vector need not be given

precisely, as we can infer the coordinate system indirectly

• Let us provide an “upright” up vector: Ԧ

𝐞𝑣𝑞 =(0,1,0)

• Provided that Ԧ

𝐞𝑣𝑞 is not parallel to Ԧ 𝐞𝑔𝑠𝑝𝑜𝑢:

𝐯 = Ԧ 𝐞𝑔𝑠𝑝𝑜𝑢 × Ԧ 𝐞𝑣𝑞, ෝ 𝐯 = 𝐯/ 𝐯 ෝ 𝐱 = − Ԧ 𝐞𝑔𝑠𝑝𝑜𝑢/ Ԧ 𝐞𝑔𝑠𝑝𝑜𝑢 ො 𝐰 = ෝ 𝐱 × ෝ 𝐯

Ԧ 𝐞𝑣𝑞 ො 𝐰 ෝ 𝐯 ෝ 𝐱 Ԧ 𝐞𝑔𝑠𝑝𝑜𝑢

WCSECS: Version B (“Look At”) (4)

• We can use the derived local camera coordinate

system to define the change of coordinates transformation (see 3D Transformations):

𝐪𝐹𝐷𝑇 = 𝑣𝑦 𝑣𝑧 𝑣𝑨 𝑤𝑦 𝑤𝑧 𝑤𝑨 𝑥𝑦 𝑥𝑧 𝑥𝑨 1 ∙ 𝐔−𝐏𝑑 ∙ 𝐪𝑋𝐷𝑇

WCSECS: Version B (“Look At”) (5)

• This version of the WCSECS transformation

computation is useful in cases where:

– There is a free roaming camera – The camera follows (observes) a certain target in space – The position (and target) are explicitly defined

PROJECTIONS

Projection

• Is the process of transforming 3D coordinates of

shapes to points on the viewing plane

• Viewing plane is the 2D flat surface that represents

an embedding of an image into the 3D space – We can define viewing systems where the “projection” surface is not planar (e.g. fish-eye lenses etc.)

• (Planar) projections are define by a projection

(viewing) plane and a center of projection (eye)

Taxonomy

• Two main categories:

– Parallel projections: infinite distance between CoP and viewing plane – Perspective projections: Finite distance between CoP and viewing plane

Where do We Perform the Projections?

• Since in projections we “collapse” a 3D shape onto a

2D surface, we essentially want to loose one coordinate (say the depth z)

• Therefore, it is convenient to perform the projection

when shapes are expressed in the ECS

Orthographic Projection (1)

• The simplest projection:
• Collapse the coordinates on plane parallel to xy at

z=d (usually 0)

𝑧′ = 𝑧 𝑦′ = 𝑦

ECS y x z

𝐪 = (𝑦, 𝑧, 𝑨) 𝐪′ = (𝑦′, 𝑧′, 𝑒) 𝑒

𝑨′ = 𝑒

𝑨 = 𝑒 (view plane)

Orthographic Projection (2)

• Very simple matrix representation
• Note that the rank of the matrix is less than its

dimension: This not a reversible transformation!

– This is also intuitively justified since we “loose” all information about depth 𝐐𝑃𝑆𝑈𝐼𝑃 = 1 1 𝑒 1

The Pinhole Camera Model

• It is an ideal camera (i.e. cannot exist in practice)
• It is the simplest modeling of a camera:

photographic Image sensor For simplicity, graphics use a “front” symmetrical projection plane

The Perspective Projection

• From similar triangles, we have:

𝑧′ = 𝑒 ∙ 𝑧 𝑨 𝑦′ = 𝑒 ∙ 𝑦 𝑨

ECS y x z

𝐪 = (𝑦, 𝑧, 𝑨) 𝐪′ = (𝑦′, 𝑧′, 𝑒) 𝑧′ 𝑧 𝑒 𝑨

𝑨′ = 𝑒

𝑨 = 𝑒 (view plane)

Matrix Form of Perspective Projection

• The perspective projection is not a linear operation

(division by z) 

• It cannot be completely represented by a linear
• perator such as a matrix multiplication

𝐐𝑄𝐹𝑆 = 𝑒 𝑒 𝑒 1 𝐐𝑄𝐹𝑆 ∙ 𝐪𝑋𝐷𝑇 = 𝑦 ∙ 𝑒 𝑧 ∙ 𝑒 𝑨 ∙ 𝑒 𝑨 𝑦 ∙ 𝑒 𝑧 ∙ 𝑒 𝑨 ∙ 𝑒 𝑨 /𝑨 = 𝑦 ∙ 𝑒/𝑨 𝑧 ∙ 𝑒/𝑨 𝑒 1

Requires a division by the w coordinate to rectify the homogeneous coordinates

Properties of the Perspective Projection

• Lines are projected to lines
• Distances are not preserved
• Angles between lines are not preserved unless lines

are parallel to the view plane

• Perspective foreshortening: The size of the

projected shape is inversely proportional to the distance to the plane

The Impact of Focal Distance d

What Happens After Projection? (1)

• Coordinates are transformed to a “post-projective”

space

Y X ECS Y X Post-projective space

What Happens After Projection? (2)

• Remember also that “depth” is for now collapsed to

the focal distance

• How then are we going to use the projected

coordinates to perform “depth” sorting in order to remove hidden surfaces?

• Also, how do we define the extents of what we can

see?

Preserving the Depth

• Regardless of what the projection is, we also retain

the transformed z values

• For numerical stability, representation accuracy and

plausibility of displayed image, we limit the z-range

• 𝑜 ≤ 𝑨 ≤ 𝑔,

– 𝑜=near clipping value, – 𝑔=far clipping value,

• The boundaries (line segments) of the image, form

planes in space:

• The intersection of the visible subspaces, defines

what we can see inside a view frustum

The View Frustum

Z X Y

The Clipping Volume (1)

• The viewing frustum, forms a clipping volume
• It defines which parts of the 3D world are discarded, i.e. do

not contribute to the final rendering of the image

• For many rendering architectures, this is a closed volume

(capped by the far plane)

Right clipping plane Near clipping plane Orthographic Clipping volume Perspective Clipping volume

The Clipping Volume (2)

• After projection, the contents of the clipping volume

are warped to match a rectangular paralepiped

• This post-projective volume is usually considered

normalized and its local coordinate system is called Canonical Screen Space (CSS)

• The respective device coordinates are also called

Normalized Device Coordinates (NDC)

Orthographic Projection Revisited (1)

• Let us now create an orthographic projection that

transforms a specific clipping box volume (left, right, bottom, top, near, far) to CSS:

Z X Y

Orthographic Projection Revisited (2)

• A simple translation  scaling transformation can

warp the clipping volume into NDC

Z X Y

(-1,-1,-1) (1, 1, 1) Notice the change of handedness here: (-1 corresponds to “near”, while “far” is 1)

Orthographic Projection Revisited (3)

Perspective Projection Revisited (1)

• We want a similar transformation to warp the

contents of the perspective frustum into a normalized cube space (CSS)

• Let us now see what happens to geometry when the

Cartesian coordinates are perspectively projected (warped) after the transformation:

Perspective Projection Revisited (2)

• In perspective projection, the clipping space is a

capped pyramid (frustum)

Perspective Projection Revisited (3)

• We still need to perform the perspective division
• We also need to retain the depth information
• Depth must obey the same transformation (division

by z)  retain straight lines

• So it must be of the general form: zs=A+B/ze
• Solving A and B for the boundary conditions:

f=A+B/f and n=A+B/n:

• A=n+f
• B=-nf 
• zs=n+f-nf/ze

Perspective Projection Revisited (4)

• zs=n+f-nf/ze

Perspective Projection Revisited (5)

Viewing frustum Post-projective (NDC) space

Perspective Projection Revisited (6)

• Next, we must normalize the result to bring it to the

CSS coordinates:

Perspective Projection Revisited (7)

• Of course, we still need to divide with the w

coordinate after the matrix multiplication

Extended Perspective Projection (1)

• In general, the frustum

axis is not aligned with the viewing direction

• To bring this frustum to

the CSS normalized volume, we must first skew it

Extended Perspective Projection (2)

• Why do we need an off-axis projection?

Stereo Multi-view rendering Planar reflections

Extended Perspective Projection (3)

• The center of the near and

far cap must coincide with the z axis

• Therefore, using the z-based

shear transformation:

• We require:

𝑐0 + 𝑢𝑝 2 + 𝐶𝑜𝑝 = 0 𝑚0 + 𝑠𝑝 2 + 𝐵𝑜𝑝 = 0

Perspective: Putting Everything Together (1)

• The final extended perspective transformation matrix:

Contributors

• Georgios Papaioannou
• Sources:

– T. Theoharis, G. Papaioannou, N. Platis, N. M. Patrikalakis, Graphics & Visualization: Principles and Algorithms, CRC Press